Some characterizations of generalized Noetherian rings induced by a multiplicatively closed set

Let Formula: see text be a multiplicatively closed set of a commutative ring Formula: see text. In this paper, Formula: see text-Noetherian rings, i.e., the rings satisfying that every Formula: see text-ideal is finitely generated, are studied. In detail, we provide the (Generalized) Principal Ideal Theorem, Hilbert Basis Theorem, Krull-Akizuki Theorem, Special Chinese Remainder Theorem, and Krull Intersection Theorem for Formula: see text-Noetherian rings. Then Formula: see text-injective modules are discussed. An Formula: see text-module Formula: see text is called Formula: see text-injective if for any Formula: see text-ideal Formula: see text of Formula: see text, every homomorphism Formula: see text can be extended to Formula: see text. The Baer’s Criterion and Faith Theorem for Formula: see text-injective modules are provided. And the Cartan-Eilenberg-Bass Theorem for Formula: see text-Noetherian rings is given in terms of Formula: see text-injective modules.